Further Mathematics - Senior Secondary 1 - Trigonometric Ratios of Special Angles

Trigonometric Ratios of Special Angles

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TERM: 1ST TERM

WEEK 9
Class: Senior Secondary School 1
Age: 15 years
Duration: 40 minutes of 4 periods
Subject: Further Mathematics
Topic: Trigonometric Ratios of Special Angles
Focus:
i. Trigonometric Ratios of 30°, 45°, and 60°
ii. Application of Trigonometric Ratios of 30°, 45°, and 60° to solve problems without the use of tables

SPECIFIC OBJECTIVES:
By the end of the lesson, students should be able to:

  1. Derive the trigonometric ratios of 30°, 45°, and 60°.
  2. Recall the exact values of sine, cosine, and tangent for 30°, 45°, and 60°.
  3. Apply the trigonometric ratios of special angles to solve simple right-angled triangle problems.
  4. Understand and interpret right-angled triangles with respect to the special angles.

INSTRUCTIONAL TECHNIQUES:
• Guided demonstration
• Question and answer
• Discussion
• Diagram interpretation
• Practice exercises

INSTRUCTIONAL MATERIALS:
• Whiteboard and markers
• Diagrams of right-angled triangles with 30°, 45°, and 60°
• Rulers, protractors, and set squares
• Worksheets for derivation and application practice

PERIOD 1 & 2: Derivation of Trigonometric Ratios of Special Angles

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the topic and explains that some angles have exact trigonometric values which can be derived geometrically.

Students listen and respond to questions.

Step 2 - Triangle for 45°

Constructs an isosceles right-angled triangle (45°-45°-90°) and derives sin 45°, cos 45°, and tan 45° using Pythagoras’ Theorem.

Students follow the construction and derive the ratios.

Step 3 - Triangle for 30° and 60°

Draws an equilateral triangle and splits it into two right-angled triangles to derive 30° and 60° angles. From this, derives sin, cos, and tan for 30° and 60°.

Students observe and copy the derivation in their notebooks.

Step 4 - Real-World Connection

Discusses practical uses of these exact values in architecture and design.

Students suggest other real-world applications.

NOTE ON BOARD:
Trigonometric Ratios of Special Angles:

  • sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3
  • sin 45° = 1/√2, cos 45° = 1/√2, tan 45° = 1
  • sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3

EVALUATION (5 exercises):

  1. State the trigonometric ratios of 30°.
  2. What is the value of cos 45°?
  3. Derive tan 60° using a triangle.
  4. What triangle is used to derive the 30° and 60° angles?
  5. Why do we not need a calculator or table for these angles?

CLASSWORK (5 questions):

  1. Derive sin 45° using a right-angled triangle.
  2. Derive cos 30° using a geometric figure.
  3. Derive tan 30° and express it in simplest form.
  4. Explain how to obtain a 30°-60°-90° triangle.
  5. State the value of sin 60° without using a calculator.

ASSIGNMENT (5 tasks):

  1. Draw a right-angled triangle with 45° and derive the trigonometric ratios.
  2. Explain how the trigonometric values of 30° are derived.
  3. Why does tan 45° = 1?
  4. Derive cos 60° from a triangle.
  5. List one practical situation where exact trigonometric values are used.

 

PERIOD 3 & 4: Application of Trigonometric Ratios of Special Angles

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Recaps the exact trigonometric values. Introduces problems involving right-angled triangles where special angles are used.

Students recall values and prepare for problem-solving.

Step 2 - Sample Problem 1

Solves a triangle problem involving a 30° angle and a known side using trigonometric ratios.

Students follow the steps and solve along with the teacher.

Step 3 - Sample Problem 2

Solves another problem involving a 60° triangle and calculates the unknown side or angle.

Students take notes and try the process.

Step 4 - Practice & Guidance

Distributes worksheets for practice questions. Guides students as they solve using exact trigonometric values.

Students work independently or in pairs and ask questions when stuck.

NOTE ON BOARD:
To solve triangle problems using special angles:

  • Identify known angle(s) and side(s).
  • Use appropriate trigonometric ratio: sin, cos, or tan.
  • Substitute the known values and solve for the unknown.
    Examples worked on the board.

EVALUATION (5 exercises):

  1. A right-angled triangle has a 30° angle and a hypotenuse of 10 cm. Find the opposite side.
  2. In a triangle with angle 60° and adjacent side of 6 cm, find the hypotenuse.
  3. Find the height of a triangle with angle 45° and hypotenuse 7√2 cm.
  4. What is the length of the opposite side in a 30° triangle with adjacent side 5 cm?
  5. Use tan 60° to find the height of a pole if the base is 4 m away.

CLASSWORK (5 questions):

  1. Solve for the opposite side in a triangle with angle 45° and hypotenuse of 10 cm.
  2. Find the adjacent side in a triangle with angle 60° and hypotenuse of 12 cm.
  3. Calculate the height of a triangle with 30° and adjacent side 7 cm.
  4. What is the length of the hypotenuse in a 45° triangle if one side is 5 cm?
  5. Use sin 60° to find the opposite side if the hypotenuse is 10 cm.

ASSIGNMENT (5 tasks):

  1. Solve a triangle with angle 30° and hypotenuse 14 cm.
  2. A ladder leans against a wall making a 60° angle with the ground. If the ladder is 8 m long, how high does it reach?
  3. Draw and solve a triangle using 45° angle and a known side.
  4. Explain when it is better to use sine over tangent.

Create your own triangle problem using any special angle and solve it.